Subgroups — #69884

Order 384.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#812	C₂	1	2	yes	maximal
3	#4466	C₂	1	4	yes	maximal, complemented
4	#2110	—	3	1	no	maximal, Sylow
5	C₂×C₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₃×#117	C₄	1	1	yes	characteristic
8	C₃×#97	C₂²	1	2	yes	—
9	C₃×C₄×C₈	C₄	1	2	yes	—
10	#812	C₄	1	4	yes	complemented
11	#812	C₂²	1	4	yes	—
12	C₂×C₃⋊C₁₆	C₄	1	4	yes	complemented
13	C₂×M₃₂	—	3	2	no	—
14	#405	D₃	1	1	yes	characteristic
15	#347	—	3	4	no	—
16	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₃×M₁₆	—	2	2	no	—
18	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
23	C₃×C₄⋊C₄	—	2	1	no	—
24	C₃×C₂²⋊C₄	—	2	1	no	—
25	C₃×C₄²	C₂×C₄	1	2	yes	—
26	C₃⋊C₁₆	C₂×C₄	1	8	yes	—
27	C₂×M₁₆	Dic₃	1	1	yes	characteristic
28	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
29	#117	Dic₃	1	1	yes	characteristic
30	M₃₂	—	3	4	no	—
31	M₃₂	—	3	4	no	—
32	C₂×C₁₆	—	3	4	no	—
33	#97	C₂×D₃	1	2	yes	—
34	C₄×C₈	Dic₃	1	2	yes	—
35	C₂³×C₃	M₁₆	1	1	yes	characteristic
36	C₂×C₃×C₄	—	2	2	no	—
37	C₂×C₃×C₄	C₄²	1	2	yes	—
38	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
39	C₂×C₃×C₄	M₁₆	1	2	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₃×C₈	—	2	2	no	—
42	C₃×C₈	C₄²	1	1	yes	characteristic
43	C₃×C₈	C₄²	1	1	yes	characteristic
44	C₃×C₈	C₂²×C₄	1	2	yes	—
45	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
46	M₁₆	—	2	2	no	—
47	C₂×C₈	C₄×D₃	1	2	yes	—
48	C₂×C₈	C₄×D₃	1	2	yes	—
49	C₂×C₈	C₂×Dic₃	1	2	yes	—
50	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
52	C₄⋊C₄	—	2	1	no	—
53	C₂²⋊C₄	—	2	1	no	—
54	C₄²	C₂×Dic₃	1	2	yes	—
55	C₁₆	—	3	8	no	—
56	C₂²×C₃	—	4	1	no	—
57	C₂²×C₃	#97	1	2	yes	—
58	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
59	C₃×C₄	—	4	2	no	—
60	C₃×C₄	#97	1	2	yes	—
61	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
62	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
63	C₂³	#208	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	#208	1	1	yes	characteristic
66	C₂×C₄	#208	1	2	yes	—
67	C₂×C₄	C₄×Dic₃	1	2	yes	—
68	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	C₄×Dic₃	1	1	yes	characteristic
71	C₈	C₄×Dic₃	1	1	yes	characteristic
72	C₈	C₂×C₄×D₃	1	2	yes	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	2	2	no	—
75	C₂×C₃	C₂×#97	1	1	yes	characteristic
76	C₂²	—	4	1	no	—
77	C₂²	#803	1	2	yes	—
78	C₂²	C₂×#208	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	#803	1	2	yes	—
81	C₄	C₂×#208	1	1	yes	characteristic
82	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
83	C₃	#2110	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#803	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₃
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#405	C₂
socle	C₂×C₃	C₂×#97
prime radical	C₂³×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#405	D₃
hypocenter	C₃	#2110
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₈ ◁ #405

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #4466	C₂	semidirect
10. #812	C₄	semidirect
12. C₂×C₃⋊C₁₆	C₄	semidirect
83. C₃	#2110	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#405	2
C₂×#812	2
#4466	2
#2110	3